The Typical Structure of Intersecting Families

When t = 1, we simply say that the family is intersecting. Consider the following example. Fix a t-set, say I ⊆ [n], and values {xi : i ∈ I}. If for every σ ∈ F and i ∈ I σ(i) = xi, then F is clearly t-intersecting. Furthermore, we say that F is a trivial t-intersecting family of permutations. Note that the size of this family is at most (n − t)!. Ellis, Friedgut, and Pilpel [5] show that for n sufficiently large with respect to t a t-intersecting family of permutations, say F , has size at most (n − t)! with equality only if F is trivial. We determine the typical structure of t-intersecting families, extending these results to show that almost all t-intersecting families are trivial. We also obtain sparse analogues of these extremal results, showing that they hold in random settings.